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A new modulus of smoothness based on the Euler angles is introduced on theunit sphere and is shown to satisfy all the usual characteristic properties ofmoduli of smoothness, including direct and inverse theorem for the bestapproximation by polynomials and its equivalence to a $K$-functional, definedvia partial derivatives in Euler angles. The set of results on the moduli onthe sphere serves as a basis for defining new moduli of smoothness and theircorresponding $K$-functionals on the unit ball, which are used to characterizethe best approximation by polynomials on the ball.

Moduli of Smoothness and Approximation on the Unit Sphere and the Unit Ball